The Secure Metric Dimension of the Globe Graph and the Flag Graph

Let G = (V, E) be a connected, basic, and finite graph. A subset <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>T</a:mi><a:mo>=</a:mo><a:mfenced open="{" close="}" separators="|"><a:mrow><a:msub><a:mrow><a:mi>u</a:mi></a:mrow><a:mrow><a:mn>1</a:mn></a:mrow></a:msub><a:mo>,</a:mo><a:msub><a:mrow><a:mi>u</a:mi></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:msub><a:mo>,</a:mo><a:mo>…</a:mo><a:mo>,</a:mo><a:msub><a:mrow><a:mi>u</a:mi></a:mrow><a:mrow><a:mi>k</a:mi></a:mrow></a:msub></a:mrow></a:mfenced></a:math> of V(G) is said to be a resolving set if for any y ∈ V(G), the code of y with regards to T, represented by <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"><f:msub><f:mrow><f:mi>C</f:mi></f:mrow><f:mrow><f:mi>T</f:mi></f:mrow></f:msub><f:mfenced open="(" close=")" separators="|"><f:mrow><f:mi>y</f:mi></f:mrow></f:mfenced></f:math> , which is defined as <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M3"><k:msub><k:mrow><k:mi>C</k:mi></k:mrow><k:mrow><k:mi>T</k:mi></k:mrow></k:msub><k:mfenced open="(" close=")" separators="|"><k:mrow><k:mi>y</k:mi></k:mrow></k:mfenced><k:mo>=</k:mo><k:mi>d</k:mi><k:mfenced open="(" close=")" separators="|"><k:mrow><k:msub><k:mrow><k:mi>u</k:mi></k:mrow><k:mrow><k:mn>1</k:mn></k:mrow></k:msub><k:mo>,</k:mo><k:mi>y</k:mi></k:mrow></k:mfenced><k:mo>,</k:mo><k:mi>d</k:mi><k:mfenced open="(" close=")" separators="|"><k:mrow><k:msub><k:mrow><k:mi>u</k:mi></k:mrow><k:mrow><k:mn>2</k:mn></k:mrow></k:msub><k:mo>,</k:mo><k:mi>y</k:mi></k:mrow></k:mfenced><k:mo>,</k:mo><k:mo>…</k:mo><k:mo>,</k:mo><k:mi>d</k:mi><k:mrow><k:mfenced open="(" close=")" separators="|"><k:mrow><k:msub><k:mrow><k:mi>u</k:mi></k:mrow><k:mrow><k:mi>k</k:mi></k:mrow></k:msub><k:mo>,</k:mo><k:mi>y</k:mi></k:mrow></k:mfenced></k:mrow></k:math> , is different for various y. The dimension of G is the smallest cardinality of a resolving set and is denoted by dim(G). If, for any t ∈ V – S, there exists r ∈ S such that <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" id="M4"><y:mfenced open="(" close=")" separators="|"><y:mrow><y:mi>S</y:mi><y:mo>–</y:mo><y:mrow><y:mfenced open="{" close="}" separators="|"><y:mrow><y:mi>r</y:mi></y:mrow></y:mfenced></y:mrow></y:mrow></y:mfenced><y:mo>∪</y:mo><y:mfenced open="{" close="}" separators="|"><y:mrow><y:mi>t</y:mi></y:mrow></y:mfenced></y:math> is a resolving set, then the resolving set S is secure. The secure metric dimension of 𝐺 is the cardinal number of the minimum secure resolving set. Determining the secure metric dimension of any given graph is an NP-complete problem. In addition, there are several uses for the metric dimension in a variety of fields, including image processing, pattern recognition, network discovery and verification, geographic routing protocols, and combinatorial optimization. In this paper, we determine the secure metric dimension of special graphs such as a globe graph <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" id="M5"><jb:mi>G</jb:mi><jb:msub><jb:mrow><jb:mi>l</jb:mi></jb:mrow><jb:mrow><jb:mi>n</jb:mi></jb:mrow></jb:msub></jb:math> , flag graph <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" id="M6"><lb:mi>F</lb:mi><lb:msub><lb:mrow><lb:mi>l</lb:mi></lb:mrow><lb:mrow><lb:mi>n</lb:mi></lb:mrow></lb:msub></lb:math> , H- graph of path <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" id="M7"><nb:msub><nb:mrow><nb:mi>P</nb:mi></nb:mrow><nb:mrow><nb:mi>n</nb:mi></nb:mrow></nb:msub></nb:math> , a bistar graph <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" id="M8"><pb:msubsup><pb:mrow><pb:mi>B</pb:mi></pb:mrow><pb:mrow><pb:mi>n</pb:mi><pb:mo>,</pb:mo><pb:mi>n</pb:mi></pb:mrow><pb:mrow><pb:mn>2</pb:mn></pb:mrow></pb:msubsup></pb:math> , and tadpole graph <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" id="M9"><rb:msub><rb:mrow><rb:mi>T</rb:mi></rb:mrow><rb:mrow><rb:mn>3</rb:mn><rb:mo>,</rb:mo><rb:mi>m</rb:mi></rb:mrow></rb:msub></rb:math> . Finally, we derive the explicit formulas for the secure metric dimension of tadpole graph <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" id="M10"><tb:msub><tb:mrow><tb:mi>T</tb:mi></tb:mrow><tb:mrow><tb:mi>n</tb:mi><tb:mo>,</tb:mo><tb:mi>m</tb:mi></tb:mrow></tb:msub></tb:math> , subdivision of tadpole graph <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" id="M11"><vb:mi>S</vb:mi><vb:mfenced open="(" close=")" separators="|"><vb:mrow><vb:msub><vb:mrow><vb:mi>T</vb:mi></vb:mrow><vb:mrow><vb:mn>3</vb:mn><vb:mo>,</vb:mo><vb:mi>m</vb:mi></vb:mrow></vb:msub></vb:mrow></vb:mfenced></vb:math> , and subdivision of tadpole graph <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" id="M12"><ac:mi>S</ac:mi><ac:mfenced open="(" close=")" separators="|"><ac:mrow><ac:msub><ac:mrow><ac:mi>T</ac:mi></ac:mrow><ac:mrow><ac:mi>n</ac:mi><ac:mo>,</ac:mo><ac:mi>m</ac:mi></ac:mrow></ac:msub></ac:mrow></ac:mfenced></ac:math> .